Mechanics Analysis Composite Materials Episode 10 docx

Chapter 7 ENVIRONMENTAL, SPECIAL LOADING, AND MANUFACTURING EFFECTS Properties of composite materials, as well as properties of all structural materials are affected by environmental and operational conditions. Moreover, for polymeric composites this influence is more pronounced than for conventional metal alloys because polymers are more sensitive to temperature, moisture, and time than metals. There exists also a specific feature of composites associated with the fact that they do not exist apart from composite structures and are formed while these structures are fabricated. As a result, material characteristics depend on the type and parameters of the manufacturing process, e.g., unidirectional composites made by pultrusion, hand lay-up, and filament winding can demonstrate different properties. This section of the book is concerned with the effect of environmental, loading, and manufacturing factors on mechanical properties and behavior of composites. 7. I. Temperature effects Temperature is the most important of environmental factors affecting the behavior of composite materials. First, because polymeric composites are rather sensitive to temperature and have relatively low thermal conductivity. This combination of properties allows us, on one hand, to use these materials in structures subjected to short-term heating, and on the other hand, requires us to perform analysis of these structures with due regard to temperature effects. Second, there exist composite materials, e.g., carbon-carbon and ceramic composites, that are specifically developed for the operation under intense heating and materials like mineral-fiber composites that are used to form heatproof layers and coatings. And third, fabrication of composite structures is usually accompa- nied with more or less intensive heating (e.g., for curing or carbonization), and the further cooling induces thermal stresses and strains to calculate which we need to attract equations of thermal conductivity and thermoelasticity that are discussed below. 301 302 Mechanics and analysis of composite muterials 7.1 .I, Thernial conductivity Heat flow through the unit area of the surface with normal n is linked with the temperature gradient in the n-direction by Fourier's law as where I is the thermal conductivity of the material. Temperature distribution along the n-axis is governed by the following equation: in which c and p are specific heat and density of the material and t is time. For a steady (time-independent) temperature distribution, aT/at = 0 and Eq. (7.2) yields Consider a laminated structure referred to coordinates x, z and shown in Fig. 7.1. To determine the temperature distribution along the x-axis only we should take into account that 2 does not depend on x, and assume that T does not depend on z. Using conditions T(x = 0) = TO and T(x = 1) = to find constants C1 and C? in Eq. (7.3), where n = x, we get 'X T = TO + - (T/ - To) 1 Introduce the apparent thermal conductivity of the laminate in the ;-direction, 3,, and write Eq. (7.1) for the laminate as Z Fig. 7.1. Temperature distribution in a laminate. Chapter 7. Environrncwlal, special loading, and rnanufucturing e&ts 303 The same equation can be written for the ith layer, i.e. The total heat flow through the laminate in the x-direction is k Substituting the foregoing results we arrive at (7.4) where h; = h;/h. thermal conductivity A, in accordance with the following form of Eq. (7.1): Now consider the heat transfer in the z-direction and introduce the apparent Taking n = z and I. = 1.; for zi-l < z < zi in Eq. (7.3), using step-wise integration and conditions T(z = 0) = TO? T(z = h) = G to find constants CI and C: we get for the ith layer The heat flow through the ith layer follows from Eqs. (7.1) and (7.6), i.e. Obviously, q; = 4: (see Fig. 7.1), and with due regard to Eq. (7.5) where as earlier, hi = h,/h. Thus obtained results, Eqs. (7.4) and (7.7), can be used to determine the thermal conductivity of a unidirectional composite ply. Indeed, comparing Fig. 7.1 with 304 Mechanics and analysh qf composite materials Fig. 3.34 showing the structure of the first-order ply model, we can write the following equations specifying thermal conductivity of a unidirectional ply along and across the fibers: Here Klr and AZf are thermal conductivities of the fiber in longitudinal and transverse directions (for some fibers they are different), A,,, is the corresponding charac- teristic of the matrix, and uf, urn = 1 - uf are fibers and matrix volume fractions. Conductivity coefficients in Eqs. (7.8) are analogous to elastic constants specified by Eqs. (3.76) and (3.78), and the discussion presented in Section 3.3 is valid for Eqs. (7.8) as well. Particularly, it should be noted that application of higher-order microstructural models practically does not change 21 but substantially improves 12 determined by Eqs. (7.8). Typical properties of unidirectional and fabric composites are listed in Table 7. I. Consider heat transfer in an orthotropic ply or layer in coordinate frame x, y whose axes x and y make angle 4 with the principal material coordinates XI and x2 as in Fig. 7.2. Heat flows in coordinates x, y and XI, x2 are linked by the following equations: q.x = q1 cos 4 - q2 sin 4, qJ, = q1 sin 4 + qz cos 4 . (7.9) Here, in accordance with Eq. (7.1) Table 7.1 Typical thermal conductivity and expansion coefficients of composite materials. Property Glass- Carbon- Aramid- Boron- Glass Aramid epoxy epoxy epoxy epoxy fabric-epoxy fabric-epoxy ~ Longitudinal 0.6 1 0.17 0.5 0.35 0.13 conductivity Transverse 0.4 0.6 0.1 0.3 0.35 0.13 conductivity Longitudinal 7.4 -0.3 -3.6 4.1 8 0.8 CTE 10' XI (]/"a Transverse 22.4 34 60 19.2 8 0.8 CTE IO" (1PC) 4 W/m K) (W/m K) Chapter 7. Eneirvnmental. special loading. and manufacturing effects 305 Fig. 7.2. Heat flows in coordinates x. 1’ and XI. x?. Changing variables xI, x’ for x, y with the aid of the following transformation relationships: x =XI cos 4 - x, sin 4, y = x, sin4 +x2 cos4 and substituting q1 and q2 into Eqs. (7.9) we arrive at where (7.10) can be treated as the ply thermal conductivities in coordinates x, y. Because the ply is anisotropic in these coordinates, the heat flow in the, e.g. x-direction induces the temperature gradient not only in the x-direction, but in the y-direction as well. Using Eq. (7.4) we can now determine the in-plane thermal conductivities of the laminate as where 1.,ti. are specified by Eqs. (7.10) in which 1.1.2 = and 4 = 4i. For &#J angle- ply laminates which are orthotropic, As an example, consider the composite body of a space telescope the section of which is shown in Fig. 7.3. The cylinder having diameter D = 1 m and total thickncss h = 13.52 mm consists of four layers, i.e. = 0. 306 Meclicmics und unnlwis of coniposite niaterials Fig. 7.3. A composite section of a space telescope. Courtesy of CRISM 0 ?qb$ angle-ply carbon-poxy external skin with the following parameters: @s = 20", hE = 3.5 mm, ET = 120 GPa, E; = 11 GPd, qz = 5.5 GPa, 19;~ = 0.27, = 1 W/m K, 2; = 0.6 W/m K, I - - -0.3 x 10 l/OC, = 34 x 10 ' 1/"C, 0 carbon-poxy lattice laye; (see Fig. 4.90) formed by a system of &4, helical ribs with 4, = 26", h, = 9 mm, 6,. = 4 mm, a,. = 52 mm, E, = 80 GPa, I., = 0.9 W/m K, E, = -1 x lop6 1/"C, 0 internal skin made of aramid fabric with h: = 1 mm, E: = E: = 34 GPa, G:, = 5.6 GPa, v:, = a: = a: = 0.8 x lop6 l/OC (x and 1' are the axial and the circumferential coordinates of the cylinder), 0 internal layer of aluminum foil with hf = 0.02 mm, Ef = 70 GPa, vf = 0.3, I,t = 210 W/m K, cq = 22.3 x 10 = 0.15, A: = A: = 0.13 Wjm K, I/OC. Apparent thermal conductivity of the cylinder wall can be found with the aid of Eqs. (7.10), (7.1 1) and the continuum model of the lattice layer described in Section 4.7 as Calculating yields A., = 0.64 W/m K. Thermal resistance of a unit length of this structure is 1 K rlr = - = 36.8 - . Dh Wm Chapter 7. Enl'i~JnV7enId. special lotding. and mflnufflcfitring effects 307 As known, heating gives rise to thermal strains which, being restricted, induce thermal stresses. Assume that the temperature distribution in a composite structure is known and consider the problem of thermoelasticity. Consider first a thermoelastic behavior of a unidirectional composite ply studied in Section 3.3 and shown in Fig. 3.29. The generalized Hooke's law, Eqs. (3.58) allowing for the temperature effects can be written as: CIT + El T , &?T = E? + &?. T 712T = 712 . (7.12) Here and further subscript "T" shows the strains that belong to the problem of thermoelasticity, while superscript "T" indicates temperature terms. Elastic strains cl,cz and *ill in Eqs. (7.12) are linked with stresses by Eqs. (3.58). Temperature strains. in the first approximation. can be taken as linear functions of the temperature change, Le. ~f = !.XIAT, C: = XIAT . (7.13) where 21 and E: are coefficients of thermal expansion (CTE) along and across the fibers and AT = T - Ti is the difference between the current temperature T and some initial temperature TO at which thermal strains are zero. Inverse form of Eqs. (7.12) is: T 61 El (CIT + 1?12Fzr) - El (8; + \'1282), 62 = EI(E?T + Y21EII') -E?(&; + \b1&;): TI: = Gi2Yll.r (7.14) where E1.2 = El,2/(l - Y~IV?~). To describe thermoelastic behavior of a ply, apply the first-order micromechani- cal model shown in Fig. 3.34. Because CTE (and elastic constants) of some fibers can be different in longitudinal and transverse directions generalize the first two equations of Eqs. (3.63) as: Repeating the derivation of Eqs. (3.76)-(3.79) we arrive at (7.15) 308 Mechanics and analysis of composite materials These equations generalize Eqs. (3.76)-(3.79) for the case of anisotropic fibers and specify apparent CTE of a unidirectional ply. As an example, consider a high-modulus carbon-epoxy composite tested by Rogers et al. (I 977). Microstructural parameters of the material are as follows (2' = 27°C): Efl = 41 1 GPa, Eo = 6.6 GPa, vfl = 0.06, V~L = 0.35, Efl = -1.2 x Em = 5.7 GPa, vm = 0.316, am = 45 x IO-' l/"C, of = om = 0.5. For these properties, Eqs. (7.16) yield El = 208.3 GPa, E2 = 6.5 GPa, V~I = 0.33, while experimental results are El = 208.6 GPa, E2 = 6.3 GPa, v21 = 0.33 Thus, it can be concluded that the first-order microstructural model providing proper results for longitudinal material characteristics fails to predict a2 with required accuracy. Discussion and conclusions concerning this problem and presented in Section 3.3 for elastic constants are valid for thermal expansion coefficients as well. For practical applications a1 and CI? are normally determined by experimental methods. However, in contrast to the elasticity problem, for which the knowledge of experimental elastic constants and material strength excludes the micromechanical models from consideration, for the thermoelasticity problems, these models provide us with useful information even if we know experimental thermal expansion coefficients. Indeed, consider a unidirectional ply that is subjected to uniform heating which induces only thermal strains, i.e., EIT = E:, E~T = E:, Y~~~ = 0. Then, Eqs. (7.14) yield GI = 0, Q = 0, 212 = 0. For homogeneous materials, this means, that no stresses occur under uniform heating. However, this is not the case for a composite ply. Generalizing Eqs. (3.74) that specify longitudinal stresses in the fibers and in the matrix we get 1/"C, CC~L 27.3 x lo-' 1/"C, tll -0.57 x IOp6 1/"C, CI? = 43.4 x I/"C, CII = -0.5 x I/"C, M? = 29.3 x IO-' 1/"C. where aI and a2 are specified by Eqs. (7.16). Thus, because thermal expansion coefficients of the fibers and the matrix are different from those of the material, there exist microstructural thermal stresses in the composite structural elements. These stresses are self-balanced. Indeed Chapter 7. Environmental, special loading, and manufacturing effeeis 309 Consider an orthotropic layer referred to coordinate axes x, y making angle Cp with the principal material coordinate axes (see Fig. 7.2). Using Eqs. (7.14) instead of Eqs. (4.56) and repeating the derivation of Eqs. (4.71) we arrive at where A,,l are specified by Eqs. (4.72) and the thermal terms are (7.17) (7.18) Here T ET2 = E: + VI'S;, E;l E; + VZlE, and E:, E: are determined by Eqs. (7.13). The inverse form of Eqs. (7.17) is &.rT = 8.r + e.r T 7 ETT Ev + E, T , YxyT = y.r,v + rry T . (7.19) Here, E,, E,,, and yxy are expressed in terms of stresses a,, c,,, and z.~? with Eqs. (4.73, while the thermal strains are Introducing thermal expansion coefficients in xy coordinate frame with the following equations: and using Eqs. (7.13) we get (7.21) As follows from Eqs. (7.19), in an anisotropic layer, uniform heating induces not only normal strains, but also the shear thermal strain. As can be seen in Fig. 7.4, 310 Mechanics and analysis of composite materials Fig. 7.4. Calculated (lines) and experimental (circles)dependencies of thermal expansion coefficients on the ply orientation angle for unidirectional thermoplastic carbon composite (- -, 0) and a &4 angle- ply layer (-, -). Eqs. (7.21) provide fair agreement with experimental results of Barnes et al. (1989) for composites with carbon fibers and thermoplastic matrix (broken line and light circles). angle-ply layer (see Section 4.5.1). This layer is orthotropic, and the corresponding constitutive equations of thermoelastisity have the form of Eqs. (7.17) in which A14 = A41 = 0, ,424 = A42 = 0, and AT2 = 0. The inverse form of these equations is E.,c~ = E? + E,, Consider a symmetric T T &IT = 8.r + Ex , Y.~!T = Yxj, > where E.~, E,., and y,,. are expressed in terms of stresses by Eqs. (4.128), while thermal strains are Using Eqs. (4.129), (7.13), (7.18) and (7.20), we arrive at the following expressions for apparent thermal expansion coefficients where (7.22) [...]... - under elevated temperature composite structures can exhibit significant creep deformation Mechanics and analysis of composite materials 332 P = 6MPa 0.8 ' 7 [ , , , 0 0 5 10 15 , t,Days(24Hours) 20 Fig 7.20 Dependence of the circumferential strain on time for a glass-epoxy cylindrical pressure vessel loaded in steps with internal pressure p 7.3.2 Durability Composite materials, to be applied to... structural materials are revealed in creep tests allowing us to plot the dependence of strain on time under constant stress Such diagrams are shown in Fig 7 I 1 for aramid-poxy composite described 1 2 3 4 0.6 0.4 - 0.2 t,year o 0 1 2 3 4 5 Fig 7 .10 Dependence of the normalized flexural strength on the time of aging for boron (I),carbon (2) aramid (3) and glass (4)epoxy composites Mechanics and analysis o composite. .. conditions, temperature, pressure, fiber and matrix materials, levels of porosity, and 318 Mechanics and analysis of composite materials material damage Because the stable moisture content is rarely reached in real composite structures, its current distribution through the laminate depends on the laminate structure and thickness Among the polymeric composites, the highest capacity for water absorption... plane stress state, e.g., for a unidirectional composite ply or layer referred to the principal material axes, Eqs (4.55) and (7.31) can be generalized as Mechanics and analysis of composite materials 328 Applying Laplace transformation to these equations we can reduce them to the form analogous to Hooke’s law, Eqs (4.59, Le (7.45) where For the unidirectional composite ply whose typical creep diagrams... characteristic is controlled mainly by the fibers or by the matrix The curves 316 Mechanics and analysis of composite materials T , "c Fig 7.6 Experimental dependencies of thermal strains on temperature (solid lines) for +4 angle-ply carbon-epoxy composite and the correspondinglinear approximations(broken lines) 1 - 0.8 - 0.6 - Om4 t O0 0 50 m 100 2 150 200 T , o C L Fig 7.7 Experimental dependencies of normalized... durability corresponding to stress ci Mechanics and analysis of composite materials 334 Strength criteria discussed in Chapter 6 can be generalized for the case of longterm loading if we change the static ultimate stresses entering these criteria for the corresponding long-term strength characteristics 7.3.3 Cyclic loading Consider the behavior of composite materials under the action of loads periodically... For carbon-epoxy composites this reduction is about 12%, for aramid-epoxy composites - about E%, and glass+poxy materials - about 35% After drying up, the effect of moisture usually disappears Cyclic action of temperature, moisture, and sun radiation results in material aging, i.e., in degradation of material properties in the process of material or structure storage For some polymeric composites,exposure... unidirectionalcarbon-epoxy composite on temperature presented in Fig 7.7 correspond to a carbon+poxy composite, but they are typical for polymeric unidirectional composites Longitudinal modulus and tensile strength, being controlled by the fibers, are less sensitive to temperature than longitudinal compressive strength, transverse and shear characteristics Analogous results for a more temperature sensitive thermoplastic composite. .. El82 (7.37) and one dash-pot simulating material viscous behavior obeying the Newton flow law de, dt rJv=q- Equilibrium and compatibility conditions for the model in Fig 7.16 are (7.38) Mechanics and analysis of composite materials 324 Fig 7.1 6 Three-element mechanical model Using the first of these equations and Eqs (7.37) and (7.38) we get a =E m + q -de, dt Taking into account that E2 = Ev = E -E '... resulting equation is as follows (7.41) Fig 7.17 Creep diagram corresponding to mechanical model in Fig 7.16 Fig 7.18 Relaxation diagram corresponding to mechanical model in Fig 7.16 Mechanics and analysis of composite materials 326 This first-order differential equation can be solved for E in the general case Omitting rather cumbersome transformations we arrive at the following solution: This result . and matrix materials, levels of porosity, and 318 Mechanics and analysis of composite materials material damage. Because the stable moisture content is rarely reached in real composite. (4) epoxy composites. 320 Mechanics and analysis of composite materials El 1.5 1 0.5 ,% Q, = 600 MPa 6, = 450 MPa 6, = 300 MPa 03 02 01 0 ' t, 0 200 400 600 800 100 0. 0.6 Mechanics and analysis of composite materials - - T, "c Fig. 7.6. Experimental dependencies of thermal strains on temperature (solid lines) for +4 angle-ply carbon-epoxy composite